Smallest $1\ 000$ digit number $x$ , such that $x$ , $x+2$ and $x+6$ are all prime?

There are fast sieve methods for finding prime clusters. Generally these work with chunks of some primorial size, having built acceptable residues for the cluster. E.g. mod 30 there are only 2 possible values for starting a triple (0,2,6), only 8 mod 210, 64 mod 2310, etc. The larger the cluster, the fewer acceptable residues so the faster it works. Unfortunately with a triple we don't get nearly the benefit from this method as we do with larger clusters.

Charles has some ideas and Pari/GP code for triplets in this post on Mersenneforum.

I have some code in Perl/ntheory that does a decent job of finding clusters. I'm sure there is faster software, but I haven't found anything open source. Separately some researchers currently working on this area include Sorenson, Waldvogel, and Wroblewski. I've discussed ideas with Waldvogel and we seem to be doing basically the same thing, though it is hard to compare since their work, like the others, is closed source.

For your example, searching to 10^7 took 45 seconds with no solutions found. That's only 2x faster than your code. But there is also an example script that runs these in parallel, so on a fast machine we can get some linear speedup.

After about 2 hours it found 10^999 + 5537073001.